Remember Casey and the milk comparison from the Identify and Write Equivalent Ratios Concept?

Well, Casey is pleased that she has been able to write the ratios, but wonders if these is an easier way to compare them. This is where simplifying ratios comes in.

**Pay attention in this Concept and you will learn how simplifying ratios can help us to compare and draw conclusions. Then we'll come back and help Casey compare her ratios.**

### Guidance

Sometimes, a ratio does not represent a clear comparison. If you look at one of the ratios in the practice problems you just finished you will see what I mean. The ratio of orange marbles to total marbles was 2 to 22. We can ** simplify** a ratio just as we would a fraction. Let’s look at the ratio 2 to 22 in the fraction form of the ratio.

\begin{align*}\frac{2}{22}\end{align*}

**We simplify a ratio in fraction form in the same way that we would simplify a fraction.** **We use the greatest common factor of both the numerator and the denominator. By dividing the numerator and the denominator by the GCF we can simplify the fraction.**

The GCF of both 2 and 22 is 2.

\begin{align*}\frac{2 \div 2}{22 \div 2} = \frac{1}{11}\end{align*}

**The simplest form of the ratio is 1 to 11. We can write this in three ways 1 to 11, 1:11 and** \begin{align*}\frac{1}{11}\end{align*}. **When we simplify a ratio in fraction form, we also write an equivalent form of the original ratio.**

\begin{align*}\frac{1}{11} = \frac{2}{22}\end{align*}

Simplify these ratios on your own. If the ratio is not written in fraction form, you will need to do that first.

#### Example A

\begin{align*}\frac{2}{10}\end{align*}

**Solution: \begin{align*}\frac{1}{5}\end{align*}**

#### Example B

6 to 8

**Solution: 3 to 4**

#### Example C

5:20

**Solution: 1:4**

Now let's think about Casey and the milk.

**Casey can write this comparison three different ways.**

\begin{align*}4\ \text{to}\ 2 \qquad \frac{4}{2} \qquad 4:2\end{align*}

**If Casey simplifies these ratios, what conclusions can she draw?**

**4 to 2 simplifies to 2 to 1**

\begin{align*}\frac{4}{2} &= \frac{2}{1}\\ 4 : 2 &= 2 : 1\end{align*}

**Casey concludes that there are twice as many non-organic brands as there are organic. When she shows her teacher, Ms. Gilson challenges Casey to do some research about organic brands of milk to bring to the grocery store manager. Casey rises to the challenge!!**

### Vocabulary

- Ratio
- a comparison between two quantities; can be written three different ways.

- Equivalent
- equal

- Simplify
- to make smaller

- Greatest Common Factor
- the largest number that will divide into two or more numbers evenly.

### Guided Practice

Here is one for you to try on your own.

Write the following ratio in simplest form.

\begin{align*}\frac{12}{18}\end{align*}

**Answer**

We can simplify this ratio just as we would a fraction because it is in fraction form. The greatest common factor of both 12 and 18 is 6. We divide both the numerator and the denominator by 6.

**\begin{align*}\frac{2}{3}\end{align*} is our answer.**

### Video Review

James Sousa, Example of Writing a Ratio as a Simplified Fraction

James Sousa, Another Example of Writing a Ratio as a Simplified Fraction

### Practice

Directions: Simplify each ratio. Write your answer in fraction form.

1. 2 to 4

2. 3:6

3. 5 to 15

4. 2 to 30

5. 10 to 15

6. \begin{align*}\frac{4}{6}\end{align*}

7. 3:9

8. 6:8

9. \begin{align*}\frac{2}{8}\end{align*}

10. \begin{align*}\frac{4}{16}\end{align*}

11. 10 to 12

12. 7:21

13. 12:24

14. 25 to 75

15. \begin{align*}\frac{27}{30}\end{align*}

16. \begin{align*}\frac{48}{60}\end{align*}

17. \begin{align*}\frac{18}{80}\end{align*}